Your task is to extend this function to also price European options. Ensure that you do not alter the structure of the input arguments to maintain backwards compatibility with the original version.

function price $=$ binomial$\_$pricer$($S$0,$ K$,$ r$,$ sigma, T$,$ type, NumSteps, varargin$)$

$\%$ Values American Options, both calls and puts, accommodating different

$\%$ types of dividends

$\%$INPUTS

$\%$ S$0$ : Initial Stock Price

$\%$ K : Strike Price

$\%$ r : Risk$-$free Rate

$\%$ sigma : Stock volatility

$\%$ T : Term to Maturity

$\%$ type : Option type $=$ Call $\text{'}$C$\text{'}$ or Put $\text{'}$P$\text{'}$

$\%$ NumSteps : Number of steps in binomial tree

$\%$ varargin : variable number of other input arguments.

$\%$ No dividend: enter no extra input argument.

$\%$ Continuous dividend yield

$\%$ y : $-1$ extra input which is the yield.

$\%$ Discrete dividends

$\%$ D : the dividend vector

$\%$ xdate : time to ex$-$dividend date vector $($in years$)$

$\%$OUTPUT

$\%$ price : option value

$\%\%$ compute tree parameters

dt $=$ T$/$NumSteps; $\%$length of each time step, in years

u $=$ exp$($sigma$*$sqrt$($dt$))$;

d $=1/$u;

q $=($exp$($r $*$dt$)-$ d$)/($u $-$ d$)$; $\%$risk neutral prob of up state

df $=$ exp$(-$r$*$dt$)$; $\%$discount factor

$\%\%$ Forward propagation to construct the underlying price tree

$\%$ First, preallocation

St $=$ zeros$($NumSteps $+1,$ NumSteps $+1)$; $\%$ matrix to hold underlying prices

ft $=$ zeros$($NumSteps $+1,$ NumSteps $+1)$; $\%$ matrix to hold option prices

$\%$ Construct price tree

$\%$ Go through different scenarios of dividend based on the number of

$\%$ variable input arguments

n $=$ length$($varargin$)$;

switch n

case $0\%$ No dividend

$\%$ Populate the tree with prices

St$(1,1)=$ S$0$;

for j $=2$:NumSteps $+1$

for i $=1$:j$-1$

St$($i$,$j$)=$ St$($i$,$ j$-1)*$u;

end

St$($i$+1,$j$)=$ St$($i$,$ j$-1)*$d;

end

case $1\%$ dividend yield

y $=$ varargin$\{1\}$; $\%$

$\%$ Recalculate q

q $=($exp$(($r $-$ y$)*$dt$)-$ d$)/($u $-$ d$)$;

$\%$ Populate the tree with prices

St$(1,1)=$ S$0$;

for j $=2$:NumSteps $+1$

for i $=1$:j$-1$

St$($i$,$j$)=$ St$($i$,$ j$-1)*$u;

end

St$($i$+1,$j$)=$ St$($i$,$ j$-1)*$d;

end

case $2\%$ Scenario $3$: discrete dividend

$\%$ load input variables for this scenario

D $=$ varargin$\{1\}$; $\%$dividend amount

xdate $=$ varargin$\{2\}$; $\%$dividend timing

$\%$ Remove NPV of dividends from initial price

pv$\_$D $=$ npv$($D$,$ xdate, r$)$;

St$(1,1)=$ S$0-$pv$\_$D;

$\%$ Construct price tree that excludes the dividend

for j $=2$:NumSteps $+1$

for i $=1$:j$-1$

St$($i$,$j$)=$ St$($i$,$ j$-1)*$u;

end

St$($i$+1,$j$)=$ St$($i$,$ j$-1)*$d;

end

$\%$ Then add back dividend for nodes that occur before ex div date

tau$=0$; $\%$Track time through tree steps

for j $=1$:NumSteps

$\%$ check if current time step is before the dividend

time$\_$to$\_$xdate $=$ xdate $-$ tau;

idx $=$ find$($time$\_$to$\_$xdate$>0)$;

if ~isempty$($idx$)$

time$\_$to$\_$xdate $=$ time$\_$to$\_$xdate$($idx$)$;

div $=$ D$($idx$)$;

div$\_$to$\_$addback $=$ npv$($div$,$ time$\_$to$\_$xdate, r$)$; $\%$ pv of relevant future dividends

St$($:$,$j$)=$St$($:$,$j$)+$div$\_$to$\_$addback;

end

taudt; $\%$Increase by $1$ timestep

end

end

$\%\%$ Compute option prices at terminal nodes and perform backward induction to complete the option price tree

if type $==\text{'}$C$\text{'}$

ft$($:$,$ end$)=$ max$($St$($:$,$ end$)-$ K$,0)$;

$\%$ Backward induction

for j $=$ NumSteps :$-1$:$1$

for i $=1$:j

CV $=$ df $*($q$*$ft$($i$,$ j$+1)+(1-$q$)*$ft$($i$+1,$ j$+1))$;

IV $=$ St$($i$,$j$)-$ K;

ft$($i$,$j$)=$ max$($CV$,$ IV$)$;

end

end

elseif type $==\text{'}$P$\text{'}$

ft$($:$,$ end$)=$ max$($ K $-$ St$($:$,$ end$),0)$;

$\%$ Backward induction

for j $=$ NumSteps :$-1$:$1$

for i $=1$:j

CV $=$ df $*($q$*$ft$($i$,$ j$+1)+(1-$q$)*$ft$($i$+1,$ j$+1))$;

IV $=$ K $-$ St$($i$,$j$)$;

ft$($i$,$j$)=$ max$($CV$,$ IV$)$;

end

end

else

error$(\text{'}$binomial$\_$pricer $-$ Option type must be C or P$\text{'})$;

end

price $=$ ft$(1,1)$;